Let's say we have a two dimensional MC defined on the state space $\mathbb{N}\times \mathbb{N}$ evolving as below:
$(i,j) \rightarrow (i,j+1)$ with rate $\lambda$ for all $i,j$.
$(i,j) \rightarrow (i-1,j+1)$ with rate $\alpha$ for $i > K$ $(K\ge 1)$.
$(i,j) \rightarrow (i-1,j)$ with rate $\mu$ for $i \le K$.
My intention is to calculate the mean absorbing time from state $(1,j)$ to an arbitrary state where $i=0$ (let's say, $(0,c)$), denoted by $T_{(1,j)}$.
Notice that when $i \le K$, any steps in $i,j$ do not happen at the same time. I wonder if it is possible to split the Markov chain defined on $[0,K]\times \mathbb{N}$, into two independent ones: $i \rightarrow i-1$ with rate $\mu$, and $j \rightarrow j+1$ with rate $\lambda$ and analyze them independently, then we can have $T_{(1,j)}=T_{(1,j) \rightarrow (0,c)}=T_{i=1 \rightarrow i=0}=\frac{1}{\mu}$?
